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german
3 years ago
15

Use Newton's method to estimate the requested solution of the equation. Start with given value of X0 and then give x2 as the est

imated solution.
x3 + 5x +2 = 0; x0 = -1; Find the one real solution.
Mathematics
1 answer:
levacccp [35]3 years ago
6 0

Answer:

<h2>-0.3913 </h2>

Step-by-step explanation:

Given the initial value of X0 = -1, we can determine the solution of the equation x³ + 5x +2 = 0 using the Newton's method. According to newton's approximation formula;

y = f(x_0) + f'(x_0)(x-x_0)

x_n = x_n_-_1  - \frac{f(x_n_-_1 )}{f'(x_n_-_1 )}

If x_0 = 1\\

We will iterate using the formula;

x_1 = x_0  - \frac{f(x_0 )}{f'(x_0 )}

Given f(x) = x³ + 5x +2

f(x0) = f(-1) = (-1)³ + 5(-1) +2

f(-1) = -1 -5 +2

f(-1) = -4

f'(x) = 3x²+5

f'(-1) = 3(-1)²+5

f'(-1) = 8

x_1 = -1+4/8\\x_1 = -1+0.5\\x_1 = -0.5\\\\x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}\\x_2 = -0.5 - \frac{f(-0.5)}{f'(-0.5)}

f(-0.5) = (-0.5)³ + 5(-0.5) +2

f(-0.5) = -0.125-2.5+2

f(-0.5) = -0.625

f'(-0.5) = 3(-0.5)²+5

f'(-0.5) = 3(0.25)+5

f'(-0.5) = 0.75+5

f'(-0.5) = 5.75

x_2 = -0.5 - \frac{(-0.625)}{5.75}\\x_2 = -0.5 + \frac{(0.625)}{5.75}\\x_2 = -0.5 + 0.1086957\\x_2 = -0.3913

The estimated solution is -0.3913 (to 4dp)

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Step-by-step explanation:

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The divisor (x - 1) from long division corresponds to the divisor 1 in synthetic division.  Setting up synthetic division, we get:

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